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This paper studies finite sample minimax optimal randomization schemes and estimation schemes in estimating parameters including the average treatment effect, when treatment effects are heterogeneous. A randomization scheme is a distribution over a group of permutations of a given treatment assignment vector. An estimation scheme is a joint distribution over assignment vectors, linear estimators, and permutations of assignment vectors. The key element in the minimax problem is that the worst case is over a class of distributions of the data which is invariant to a group of permutations. First, I show that given any assignment vector and any estimator, the uniform distribution over the same group of permutations, namely the complete randomization scheme, is minimax optimal. Second, under further assumptions on the class of distributions and the objective function, I show the minimax optimal estimation scheme involves completely randomizing an assignment vector, while the optimal estimator is the difference-in-means under complete invariance and a weighted average of within-block differences under a block structure, and the number of treated units is determined by the Neyman allocation.

We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a well-known result that when the source and the adversary have power constraints $ P $ and $ N $ respectively, the list decoding capacity is equal to $ \frac{1}{2}\log\frac{P}{N} $. Random spherical codes achieve constant list sizes, and the goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity. We show a reduction from arbitrary codes to spherical codes, and derive a lower bound on the list size of typical random spherical codes. We also give an upper bound on the list size achievable using nested Construction-A lattices and infinite Construction-A lattices. We then define and study a class of infinite constellations that generalize Construction-A lattices and prove upper and lower bounds for the same. Other goodness properties such as packing goodness and AWGN goodness of infinite constellations are proved along the way. Finally, we consider random lattices sampled from the Haar distribution and show that if a certain number-theoretic conjecture is true, then the list size grows as a polynomial function of the gap-to-capacity., Comment: Significantly revised

A basic assumption of the classic reinsurance model is that the distribution of the loss is precisely known. In practice, only partial information is available for the loss distribution due to the lack of data and estimation error. We study a distributionally robust reinsurance problem by minimizing the maximum Value-at-Risk (or the worst-case VaR) of the total retained loss of the insurer for all loss distributions with known mean and variance. Our model handles typical stop-loss reinsurance contracts. We show that a three-point distribution achieves the worst-case VaR of the total retained loss of the insurer, from which the closed-form solutions of the worst-case distribution and optimal deductible are obtained. Moreover, we show that the worst-case Conditional Value-at-Risk of the total retained loss of the insurer is equal to the worst-case VaR, and thus the optimal deductible is the same in both cases.

Let X k 1 k = 1 ∞ , X k 2 k = 1 ∞ , … , X k d k = 1 ∞ be d independent sequences of Bernoulli random variables with success-parameters p 1 , p 2 , … , p d respectively, where d ≥ 2 is a positive integer, and 0 p j 1 for all j = 1 , 2 , … , d . Let S j ( n ) = ∑ i = 1 n X i j = X 1 j + X 2 j + ⋯ + X n j , n = 1 , 2 , … . We declare a “rencontre” at time n , or, equivalently, say that n is a “rencontre time,” if S 1 ( n ) = S 2 ( n ) = ⋯ = S d ( n ) . We motivate and study the distribution of the first (provided it is finite) rencontre time.

The performance analysis for the MIMO-FSO systems employing EGC technology over Lognormal-Rician turbulence channels with pointing errors is important. However, the results so far are greatly limited since the PDF of Lognormal-Rician turbulence channels is not analytic, even for the SISO systems. In this paper, we propose a new method to approximate the sum of lognormal-Rician turbulence channels with Rayleigh pointing errors. Based on the developed formula, the approximate closed-form expressions of ergodic capacity, outage probability, and bit-error rate are derived. Numerical results demonstrate the accuracy of the proposed approach.

In this paper we obtain explicit estimates and existence results on the number of F q -rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.

A statistical model is a class of probability distributions assumed to contain the true distribution generating the data. In parametric models, the distributions are indexed by a finite-dimensional parameter characterizing the scientific question of interest. Semiparametric models describe the distributions in terms of a finite-dimensional parameter and an infinite-dimensional component, offering more flexibility. Ordinarily, the statistical model represents distributions for the full data intended to be collected. When elements of these full data are missing, the goal is to make valid inference on the full-data-model parameter using the observed data. In a series of fundamental works, Robins, Rotnitzky, and colleagues derived the class of observed-data estimators under a semiparametric model assuming that the missingness mechanism is at random, which leads to practical, robust methodology for many familiar data-analytic challenges. This article reviews semiparametric theory and the key steps in this derivation.

We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution that is the solution of the Bellman equation, thus giving the optimal control for the problem. In addition, we study the convergence of the limit differential equation to the stationary solution. As a by-product of our analysis, we obtain the limiting behavior of single-layer neural networks when trained on independent and identically distributed data with stochastic gradient descent under the widely used Xavier initialization.

Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f ′ ′ + P ( z ) f = 0 , where P ( z ) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E. Hille’s 1969 book Lectures on Ordinary Differential Equations are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille’s book.

The subject of this paper is the problem of optimal stopping of a sequence of independent and identically distributed random variables with unknown distribution. We propose a stopping rule that is based on relative ranks and study its performance as measured by the maximal relative regret over suitable nonparametric classes of distributions. It is shown that the proposed rule is first-order asymptotically optimal and nearly rate optimal in terms of the rate at which the relative regret converges to zero. We also develop a general method for numerical solution of sequential stopping problems with no distributional information and use it in order to implement the proposed stopping rule. Some numerical experiments illustrating performance of the rule are presented as well.

In this paper, a flexible version of the Fréchet distribution called the extended Weibull–Fréchet (EWFr) distribution is proposed. Its failure rate has a decreasing shape, an increasing shape, and an upside-down bathtub shape. Its density function can be a symmetric shape, an asymmetric shape, a reversed-J shape and J shape. Some mathematical properties of the EWFr distribution are explored. The EWFr parameters are estimated using several frequentist estimation approaches. The performance of these methods is addressed using detailed simulations. Furthermore, the best approach for estimating the EWFr parameters is determined based on partial and overall ranks. Finally, the performance of the EWFr distribution is studied using two real-life datasets from the medicine and engineering sciences. The EWFr distribution provides a superior fit over other competing Fréchet distributions such as the exponentiated-Fréchet, beta-Fréchet, Lomax–Fréchet, and Kumaraswamy Marshall–Olkin Fréchet.

It is of great importance for physicists and engineers to assess a lifetime distribution of a series-parallel system when its components' lifetimes are subject to a finite mixture of distributions. The present article addresses this problem by introducing a new distribution called "Poisson-geometric-Lomax distribution". Important properties of the proposed distribution are discussed. When the stress is an increasing nonlinear function of time, the progressive-stress model is considered and the inverse power-law model has suggested a relationship between the stress and the scale parameter of the proposed distribution. Based on the progressive type-II censoring with binomial removals, estimation of the included parameters is discussed using maximum likelihood and Bayes methods. An example, based on two real data sets, demonstrates the superiority of the proposed distribution over some other known distributions. To compare the performance of the implemented estimation methods, a simulation study is carried out. Finally, some concluding remarks followed by certain features and motivations to the proposed distribution are presented.

We investigate distribution of eigenvalues of growing size Toeplitz matrices [ a n + k − j ] 1 ≤ j , k ≤ n as n → ∞ , when the entries { a j } are “smooth” in the sense, for example, that for some α > 0 , a j − 1 a j + 1 a j 2 = 1 − 1 α j ( 1 + o ( 1 ) ) , j → ∞ . Typically they are Maclaurin series coefficients of an entire function. We establish that when suitably scaled, the eigenvalue counting measures have limiting support on [ 0 , 1 ] , and under mild additional smoothness conditions, the universal scaled and weighted limit distribution is | π log ⁡ t | − 1 / 2 d t on [ 0 , 1 ] .

The aim of this paper is to highlight the use of Zero inflated Poisson distribution and Hurdle Poisson distribution to improve goodness of fit of data with excess zeros. Claim frequency data were used for the Singapore motor insurance database available on the internet. The data are tested for the detection of excess zeros. Parameters of Zero inflated Poisson distribution are estimated using method of moments and method of maximum likelihood. Mean and variance formulas are derived for the distribution of Hurdle Poisson and estimators of method of moment and maximum likelihood of unknown parameters are derived. Parameters of Hurdle Poisson are estimated. The data are modeled using Poisson distribution, Zero inflated Poisson distribution and Hurdle Poisson distribution. Goodness of fit are tested using Chi Square test. The best distribution in the study is selected using AIC & BIC criteria. Hurdle Poisson is the best distribution for modeling the data.

This paper is an addition to the series of modification and improvement of the extant distributions which enable them to analyze the new emerging situations efficiently. We develop a new lifetime distribution by generalizing the Erlang truncated exponential distribution using the Topp Leone family of distributions. A comprehensive account of mathematical characteristics such as quantile function, moments, probability weighted moments, moment generating function, and probability generating function of the proposed distribution is presented. Some reliability measures such as hazard rate function, residual life function, and reversed residual life function are also provided. Several entropy measures including Reyni entropy, Tsallis entropy, cumulative Tsallis entropy, and dynamic cumulative Tsallis entropy are obtained. Besides, the extropy, residual extropy, and cumulative residual extropy are explored. The unknown parameters of the proposed distribution are estimated by using the maximum likelihood method. The stability of the model parameters is examined through the simulation study. The application of our proposed distribution is explained through three real-life examples and its performance is illustrated through its comparison with the competent existing distributions.

A fundamental problem in coding theory is to find the values of A ( n , d ) , i.e., the A ( n , d ) is the size of the maximum binary code set of length n and minimum Hamming distance d . And A ( n , d ) can also be viewed as the size of maximum independent set of d − 1 t h -power of the n -dimensional hypercube Q n . In order to further understand and study this problem, we study the distribution of the maximum clique and the structure of the point of Q n d for d ≤ 5 .

The concept of reference change values (RCVs) for diagnosis and monitoring of diseases has become well established. Several models habe been developed, e. g. one assuming a normal distribution and another one for a log-normal distribution. RCV values calculated for some measurands with both models are compared with each other and led to similar results. A few examples led to RCV values which are not plausible for diagnostic purposes. Although statistical concepts of RCV values are well established, their clinical relevance remains questionable at least for some measurands. Studies with clinicians are required whether RCVs are of practical usefulness.

For a system (or unit) whose lifetime is measured by the number cycles, according to the discrete time age replacement policy, it is replaced preventively after n cycles or correctively at failure, whichever occurs first. In this paper, discrete time age replacement policy is revisited when the lifetime of the system is modeled by a discrete phase-type distribution. In particular, the necessary conditions for the unique and finite replacement cycle which minimizes the expected cost per unit of time are obtained. The necessary conditions are mainly based on the behavior of the hazard rate. The results are illustrated for some special discrete phase-type lifetime distributions. Computational results are also presented for the optimal replacement cycle under specific real life setups.

The fuzzy time series forecasting model is a powerful tool in forecasting the time series data. The nature of the fuzzy set exhibits its role in handling the uncertainty of the data. The intuitionistic fuzzy set (IFS) is a generalization of a fuzzy set that makes the forecasting process more precise and accurate. This paper proposes a new fuzzy forecasting model based on IFS via the de-i-fuzzification approach, namely equal distribution of hesitancy. The proposed model consists of four main parts; the fuzzification of historical data; the establishment of the IFS; the de-i-fuzzification; and the defuzzification. For the fuzzification, the historical data is partitioned into 14 intervals using the frequency density-based method and trapezoidal fuzzy numbers are used to fuzzify the data. The data are then converted into IFS. The data in IFS form is reduced to fuzzy set using equally distributed with the degree of hesitancy approach. The arithmetic rules based on centroid defuzzification is used to calculate the forecasted output. The proposed model shows a better performance than the existing forecasting models based on IFS, indicating that the equal distribution of hesitancy de-i-fuzzification managed to handle the non-determinism in the forecasting with simplified procedure. In the future, an improved method will be proposed to defuzzify the IFS into crisp values without going through the de-i-fuzzification process, yet preserving the nature of IFS.

The aim of the paper is to compute projective maximum distance separable codes, -MDS of two and three dimensions with certain lengths and Hamming weight distribution from the arcs in the projective line and plane over the finite field of order twenty-five. Also, the linear codes generated by an incidence matrix of points and lines of were studied over different finite fields.

Many problems in real life exist that are full of confusion, vagueness, and ambiguity. The quantification of such issues in a scientific way is the need of time. The negative binomial distribution is an important discrete probability distribution from the account of classical probability distribution theory. The distribution was used to study the chance of kth success in n trials before n − 1 failures for crisp data. The literature lacks in dealing with the situations for interval-valued data under negative binomial distribution. In this research, the neutrosophic negative binomial distribution is proposed to generalize the classical negative binomial distribution. The generalized proposed distribution considers the indeterminacy and crisp form from interval-valued. Several properties of the proposed distribution, such as moment generating function, characteristic function, and probability generating function, are also derived. Furthermore, the derivation of reliability analysis properties such as survival, hazard rate, reversed hazard rate, cumulative hazard rate, mills ratio, and odds ratio are also presented. In addition, order statistics for the proposed distribution, including w th , joint, median, minimum, and maximum order statistics are part of the paper. The proposed distribution is discussed from the real data applications perspective by considering the different case studies. This research opens the way to deal with the problems that follow conventional conveyances and include nonprecisely determined details simultaneously.

This paper aims to solve the problem of food safety in catering O2O distribution link. We applied the system dynamics method to model the formation mechanism of food safety risk in the distribution link. The results of our experiment include identifying the risk factors that may be faced by food safety in the distribution link from five perspectives: O2O catering enterprise’s own risk, logistics distribution team’s distribution risk, O2O catering platform supervision risk, user-supervision risk, and government department supervision risk, and establishing a risk index evaluation system based on the Analytic Hierarchy Process. With the help of the system dynamics model, the corresponding risk formation mechanism system model flow diagram is established, and the model simulation analysis is carried out. Through this research, we concluded that we can use the risk model to understand the risks faced by different subjects so as to make targeted countermeasures.

The Dirac-[Formula: see text] distribution may be realized through sequences of convolutions, the latter being also regarded as approximation to the identity. This paper proposes the real form of pre-orthogonal adaptive Fourier decomposition (POAFD) method to realize fast approximation to the identity. It belongs to sparse representation of signals having potential applications in signal and image analysis.

A new robust Kalman filter (KF) based on mixing distribution is presented to address the filtering issue for a linear system with measurement loss (ML) and heavy-tailed measurement noise (HTMN) in this paper. A new Student’s t-inverse-Wishart-Gamma mixing distribution is derived to more rationally model the HTMN. By employing a discrete Bernoulli random variable (DBRV), the form of measurement likelihood function of double mixing distributions is converted from a weighted sum to an exponential product, and a hierarchical Gaussian state-space model (HGSSM) is therefore established. Finally, the system state, the intermediate random variables (IRVs) of the new STIWG distribution, and the DBRV are simultaneously estimated by utilizing the variational Bayesian (VB) method. Numerical example simulation experiment indicates that the proposed filter in this paper has superior performance than current algorithms in processing ML and HTMN.

International audience; In this paper, we study the performance properties of the phase II exponential chart with an unknown in-control (IC) rate parameter, used to monitor high-yield processes. The average time to signal (ATS) is used as the chart performance criterion instead of the usual average run length (ARL). Based on the IC conditional ATS (CATS) distribution, we examine the properties of both the equal-tailed and the ATS-unbiased exponential charts with estimated control limits and adjust the limits so that a nominal IC ATS performance is achieved. Two perspectives are investigated: the unconditional, under which the average of IC CATS distribution is set equal to a nominal ATS and, the conditional so that the IC CATS is set to or exceed a nominal ATS with a high probability. It is shown that the ATS-unbiased exponential chart under the conditional perspective has a better IC performance.

The paper aims at describing the findings and conclusions formulated in the analysis of the authorship factor in legal discourse. It is hypothesised that verbal structures show systemically varied distribution across legal discourse and the relevant distinctions run through the authorship categories. When it comes to the aim of the research it draws on the tradition of sociolinguistic methodology targeting issues related to language variation which follows the basic assumptions of functional grammar. From the point of view of the material covered by the analysis it contributes to the research on legal discourse and specifically on its specialised domain referred to as corporate, company or business discourse. It provides additional empirical data pointing to the non-homogeneity of the legal style and formal distinctions originating from rich contextual background. The study is conducted on the material of a custom-designed corpus of English legal texts, classified as secondary genres. Methodologically, the study makes use of the tenets of supervised search of digitalised corpora and automatic data extraction based on discrete units, subsequent identification of recurring longer contiguous and/or non-contiguous sequences, if any, built around the axis of specific verbal structures and finally qualitative comparative analysis (characterisation) of the material. The discussion presents sample data and focuses on the most salient categories, both quantitatively and qualitatively. The inductive approach confirms the formal divergencies in the communicative situation covered by the analysis. The findings encapsulate patterns and tendencies in the quantitative distribution of verbal structures depending on the authorship category. It may be concluded that authorship is a factor delineating distinctions as regards (i) the repertoire of grammatical instruments exploited (verbal structures), which contributes to the specific stylistic profile of given authors. This shows that the thesis posed is verified positively and the study shows further, more detailed distinctions running through groups of subcategories distinguished within the authorship categories specified upon the start of the research.

In this paper, we propose a new distribution, named unit log–log distribution, defined on the bounded (0,1) interval. Basic distributional properties such as model shapes, stochastic ordering, quantile function, moments, and order statistics of the newly defined unit distribution are studied. The maximum likelihood estimation method has been pointed out to estimate its model parameters. The new quantile regression model based on the proposed distribution is introduced and it has been derived estimations of its model parameters also. The Monte Carlo simulation studies have been given to see the performance of the estimation method based on the new unit distribution and its regression modeling. Applications of the newly defined distribution and its quantile regression model to real data sets show that the proposed models have better modeling abilities than competitive models. The proposed unit quantile regression model has targeted to explain linear relation between educational measurements of both OECD (Organization for Economic Co-operation and Development) countries and some non-members of OECD countries, and their Better Life Index. The existence of the significant covariates has been seen on the real data applications for the unit median response.

In this paper we give few expressions and asymptotics of ruin probabilities for a Markov modulated risk process for various regimes of a time horizon, initial reserves and a claim size distribution. We also consider few versions of the ruin time.

When a garment is given a certain amount of ease, the body shape characteristics and matching degree of air gap distribution determine the aesthetic fit of garment sculpt. The purpose of this study is to investigate the relationship between the aesthetic fit and air gap distribution of the bust and waist sections. Three factors, the ratio of the horizontal radius vector to the bust and waist sections ( HrvB, HrvW) and the location and proportion of darts ( DL, DP), have been considered in this analysis. Ten female participants with different Hrv (including HrvB and HrvW) were recruited and asked to wear 10 sample blouses with different DL and DP. Samples worn by participants are observed at different angles, and evaluated with scores by 30 experts. The air gap thickness (THA) between the garment and skin are measured by a TC2 scanner and Qualisys three-dimensional motion capture system. The results show that Hrv is the most important factor affecting the aesthetic fit, while DL and DP have significant influence on the formation of garment wrinkles. A round body shape has higher scores when wearing sample garments whose waist dart is close to the front and middle (samples L2, L3) and smaller back middle dart and front waist dart (samples P1, P2), while a flat body shape has higher scores in the opposite condition (sample L5, P5). In terms of air gap distribution, there is a significant positive correlation between HrvB and THA in the front area (132–180°). Here, HrvW was negatively correlated with THA in the back side area. The correlation coefficients are all above 0.6. The DL has a great influence on THA in the side area of the bust section and front side area of the waist section, while DP has a great influence on THA in the front and back areas of the bust section and the back side area of the waist section. In a comparative analysis, this paper proved that the smaller fluctuation of THA and the uniform ease distribution are more likely to present an aesthetic and fit dressing appearance. Although the relationship between THA and the three factors was not significant, the regularity of its distribution is accompanied by a significant change of dressing score.

The goal of this paper is to develop an optimal statistical model to analyze COVID-19 data in order to model and analyze the COVID-19 mortality rates in Somalia. Combining the log-logistic distribution and the tangent function yields the flexible extension log-logistic tangent (LLT) distribution, a new two-parameter distribution. This new distribution has a number of excellent statistical and mathematical properties, including a simple failure rate function, reliability function, and cumulative distribution function. Maximum likelihood estimation (MLE) is used to estimate the unknown parameters of the proposed distribution. A numerical and visual result of the Monte Carlo simulation is obtained to evaluate the use of the MLE method. In addition, the LLT model is compared to the well-known two-parameter, three-parameter, and four-parameter competitors. Gompertz, log-logistic, kappa, exponentiated log-logistic, Marshall–Olkin log-logistic, Kumaraswamy log-logistic, and beta log-logistic are among the competing models. Different goodness-of-fit measures are used to determine whether the LLT distribution is more useful than the competing models in COVID-19 data of mortality rate analysis.

In this paper, we present a new lifetime distribution known as the generalized inverse power Sujatha distribution. The statistical and mathematical properties of the new distribution such as the moment and moment generating function, Renyi entropy and distribution of order statistics have been derived and discussed. Also, reliability measures like survival function, hazard function, reverse hazard rate, cumulative hazard rate and odds function are discussed. Maximum likelihood estimation technique was used to estimate the parameters. However, a 95% confidence intervals were constructed for the parameters. Finally, we applied the proposed distribution to two lifetime datasets and compare its superiority over other candidate models. Results obtained indicates that the generalized inverse power Sujatha distribution outperform the other competing models.

The present study attempts to explore the distribution patterns of the valency-changing verbs from the perspective of quantitative linguistics. We took authentic spoken language data as the research materials. The corpus used in this paper is a self-built spoken English corpus containing about 21,000 words. We half-manually annotated the corpus with the help of SpaCy, a natural language processing tool. According to the annotation results and statistical data, we obtained a total of 217 valency-changing English verbs and 248 sentence components governed by them. After analysis, the current study came to the following conclusions: First, bivalent verbs are most frequent among the three types of valency-changing verbs; second, after fitting all the language data to different probability distributions, we found that the rank-frequency distributions of all the valency-changing English verbs with different numbers of obligatory arguments obey the power law, and the frequencies of bivalent valency-changing verbs obey other kinds of distributions such as the mixed Poisson distribution.

Entropy measurement plays an important role in the field of information theory. Furthermore, the estimation of entropy is an important issue in statistics and machine learning. This study estimated the Rényi and q-entropies of a power-function distribution in the presence of s outliers using classical and Bayesian procedures. In the classical method, the maximum likelihood estimators of the entropies were obtained and their performance was assessed through a numerical study. In the Bayesian method, the Bayesian estimators of the entropies under uniform and gamma priors were acquired based on different loss functions. The Bayesian estimators were computed empirically using a Monte Carlo simulation based on the Gibbs sampling algorithm. The simulated datasets were analyzed to investigate the accuracy of the estimates. The study results showed that the precision of the maximum likelihood and Bayesian estimates of both entropies improved with increasing the sample size and the number of outliers. The absolute biases and the mean squared errors of the estimates in the presence of outliers exceeded those of the corresponding estimates in the homogenous case (no-outliers). Furthermore, the Bayesian estimates of the Rényi and q-entropies under the squared error loss function were preferable to the other Bayesian estimates in a majority of the cases. Finally, analysis results of real data examples were consistent with those of the simulated data.

In real applications of artificial and intelligent decision-making systems, how to represent the knowledge involved with uncertain information is still an open issue. The negation method has great significance to address this issue from another perspective. However, it has the limitation that can be used only for the negation of the probability distribution. In this article, therefore, we propose a generalized model of the traditional one, so that it can have more powerful capability to represent the knowledge, and uncertainty measure. In particular, we first define a vector representation of complex-valued distribution. Then, an entropy measure is proposed for the complex-valued distribution, called $\mathcal {X}$ entropy. In this context, a transformation function to acquire the negation of the complex-valued distribution is exploited on the basis of the newly defined $\mathcal {X}$ entropy. Afterward, the properties of this negation function are analyzed, and investigated, as well as some special cases. Finally, we study the negation function on the view from the $\mathcal {X}$ entropy. It is verified that the proposed negation method for the complex-valued distribution is a scheme with a maximal entropy.

CV. Bintang Anugerah Sukses is a company engaged in the distribution of product PT. Belfoods Indonesia. Distribution of frozen goods distributes to 12 customer agencies using one vehicle. The route distribution is determined based on the experience and subjective decisions of a salesman in charge of distributing goods. The problem will be solved using the Travelling Salesman Problem (TSP) approach, so the best route for distributing goods can be obtained that can minimum mileage and time. One of the methods applied to the traveling salesman problem is the Tabu Search algorithm. Tabu Search is one of the metaheuristic methods with the process of searching moving from one solution to the next. The algorithm uses a taboo list to store a set of solutions that have just been evaluated, the result will be adjusted first to the contents of the taboo list to see whether the solution already exists or not. If the solution already exists then the solution will not be evaluated. The calculation results obtained a minimum mileage of 53.6 km and a minimum of time 305 minutes or 5 hours 5 minutes.

The information volume of mass function based on extropy is proposed in this paper. Although the information volume of the probability distribution can be calculated by Shannon entropy, how to calculate the information of the mass function is still being explored. Recently, the concept of extropy was proposed by Lad et al. Based on extropy, the information volume of mass function is proposed in this paper. For a basic probability assignment function (BPA), if the focal elements of the frame of discernment (FOD) are all single elements, the information volume proposed in this paper is equal to the corresponding extropy. Otherwise, the information volume is greater than the corresponding extropy. Besides, when the cardinality of the FOD is identical, both the total uncertainty case and the mass function distribution of the maximum extropy have the same information volume. More precisely, the distribution of the latter can be regarded as the former obtained by decomposing the BPA once. Finally, the experiment proves that the maximum information volume increases with the increase in the cardinality of the FOD, and has the same limit value log $$_2e$$ as the maximum extropy. Some numerical examples are given to prove the nature of the information volume.

Subject. The article provides the results of meso-dynamics analysis of main twelve industries, based on monthly data for 82 Russian regions, from January 2005 till December 2020. Objectives. The purpose of the study is to address the problem of balanced and stable spatial development of Russia’s regions and Russia, which requires modeling of adequate tools and forecasting nonlinear mesodynamics. Methods. The study follows the econophysics methodology. Results. We consider additive and multiplicative interactions of regular time series components between each other and the residuals, thus expanding the scope of tools application for the variety of considered industries and their models. Using the common and new trend models, we analyze structural changes, introduce the topological measure of proximity to the neighborhood of residuals with heavy-tailed distribution, which is estimated by median values of trends and cycles for regular components. The traditional time series decomposition (by trend, cycle, seasonality, and residual) is supplemented by our unique complex of wavelet transformations, which forms the models of cycles, using auto regression. We obtained representative and time-synchronized analytical estimates of regular components of industries’ dynamics for meso- and macro-indicators of the Russian economy that have higher accuracy than the known results for the accuracy of modeling and forecasting. Conclusions. The offered methodology and tools enable a more adequate analysis of non-linear dynamics of regions’ middle-term development. They help shift to growth point identification, create the atlas of economic industrial cycles, analyze stages of bifurcations and scenario predictive planning.

We introduce the notion of an oscillatory formal distribution supported at a point. We prove that a formal distribution is given by a formal oscillatory integral if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We give an algorithm that recovers the jet of infinite order of the integral kernel of a formal oscillatory integral at the critical point from the corresponding formal distribution. We also prove that a star product $\star$ on a Poisson manifold $M$ is natural in the sense of Gutt and Rawnsley if and only if the formal distribution $f \otimes g \mapsto (f \star g)(x)$ is oscillatory for every $x \in M$., 20 pages, several typos were corrected

Random network models generated using sparse exchangeable graphs have provided a mechanism to study a wide variety of complex real-life networks. In particular, these models help with investigating power-law properties of degree distributions, number of edges, and other relevant network metrics which support the scale-free structure of networks. Previous work on such graphs imposes a marginal assumption of univariate regular variation (e.g., power-law tail) on the bivariate generating graphex function. In this paper, we study sparse exchangeable graphs generated by graphex functions which are multivariate regularly varying. We also focus on a different metric for our study: the distribution of the number of common vertices (connections) shared by a pair of vertices. The number being high for a fixed pair is an indicator of the original pair of vertices being connected. We find that the distribution of number of common connections are regularly varying as well, where the tail indices of regular variation are governed by the type of graphex function used. Our results are verified on simulated graphs by estimating the relevant tail index parameters., 21 pages, 2 figures, 3 tables